Stabilization of Markovian jump linear system over networks with random communication delay

doi:10.1016/j.automatica.2008.06.023

Automatica

Volume 45, Issue 2, February 2009, Pages 416-421

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Abstract

This paper is concerned with the stabilization problem for a networked control system with Markovian characterization. We consider the case that the random communication delays exist both in the system state and in the mode signal which are modeled as a Markov chain. The resulting closed-loop system is modeled as a Markovian jump linear system with two jumping parameters, and a necessary and sufficient condition on the existence of stabilizing controllers is established. An iterative linear matrix inequality (LMI) approach is employed to calculate a mode-dependent solution. Finally, a numerical example is given to illustrate the effectiveness of the proposed design method.

Introduction

The Markovian jump linear system (MJLS) is a linear system with randomly jumping parameters, where the jumps are modeled by the transitions of a Markov chain. Over decades, increasing effort has been devoted to discrete Markovian jump linear systems (DMJLSs) with time delay, and some important results have been reported in the existing literature ( Boukas and Liu (2001), Cao and Lam (1999), Chen, Guan, and Yu (2004), Ji, Chizeck, Feng, and Loparo (1991), Niu, Ho, and Wang (2007), Shi, Boukas, and Agarwal (1999) and Xiong, Lam, Gao, and Ho (2005) and references therein), regarding applications, stability conditions, and stabilization problems. For example, the stochastic stabilization problem for DMJLSs with state delays were investigated by Cao and Lam (1999) and Shi et al. (1999), where the results are delay-independent. Delay-dependent results were developed in Boukas and Liu (2001) and Chen et al. (2004) as well. Besides, the problem of sliding mode control (SMC) for stochastic systems with Markovian switching was studied in Niu et al. (2007). It is worth pointing out that the main control category used in Boukas and Liu (2001), Cao and Lam (1999), Chen et al. (2004), and Shi et al. (1999) is designing a control law, according to the current system mode and current system state, such that the unstable plant is stabilized without the delayed terms and remains stable in the presence of the delayed terms.

As is well known, in most practical systems, the original plant, controller, sensor and actuator are difficult to be located at the same place, and thus signals are required to be transmitted from one place to another. In modern industrial systems, these components are often connected over networks, giving rise to the so-called networked control systems (NCSs). There are many advantages in NCSs, such as low cost, reduced weight and power requirements, simple installation and maintenance, and high reliability. Thus, more and more attention has been paid to the stability and stabilization of NCSs recently (Azimi-Sadjadi (2003), Xiao and Arash hassibi (2000), Xiong and Lam (2007) and Zhang, Shi, and Chen (2005) and references therein). It should be pointed out, however, that most of the results in the existing literature are focused on NCSs where the plant is a deterministic system. To the best of the authors’ knowledge, the stability and stabilization problems for NCSs with the plant being a stochastic system have not been fully investigated to date. Especially for the case where the plant is a Markovian jump linear system, very few results related to NCSs have been available in the literature so far, which motivates the present study.

It is worth mentioning that the stochastic stabilization problem for DMJLSs with delayed input has been studied (Xiong & Lam, 2006). The main contribution of Xiong and Lam (2006) is modeling the resulting closed-loop system as a new Markovian jump linear system with extended state space. In Xiong and Lam (2006), it is assumed that a constant time delay exists in the mode signal and a time-varying delay exists in the system state. In this paper, we consider a more realistic situation as shown in Fig. 1, where the plant and controller are connected with the network, and random communication delays exist both in the system state and in the mode signal. Such a situation will cover more general cases in practical NCSs. To the authors’ best knowledge, this problem for NCSs has not been investigated in the existing literature, and unfortunately, the model setting and control category developed by Xiong and Lam (2006) cannot be directly applied to our case where the time delay in the mode signal is now randomly varying. New control techniques are needed to design a control law based upon past system information to stabilize an unstable plant. It is an important and challenging research topic, which motivates our current research. Following the work of Xiong and Lam (2006), in this paper, we will present a new method to overcome this difficulty.

In this paper, we study the stabilization of NCSs with Markovian characterization. The random communication delay is modeled as a Markov chain, and the resulting closed-loop system is modeled as a Markovian jump linear system with two jumping parameters. A stochastic stability criterion and the corresponding controller design technique are given in the form of LMIs. It should be pointed out that our method presented in this paper is not a trivial extension, but a new one with more wide application foreground.

Section snippets

Problem formulation

Consider the networked control setup in Fig. 1, where the plant is a discrete-time Markovian jump linear system defined on a complete probability space ( $Ω$ , $F$ , $P$ ): $\begin{matrix} x (k + 1) = A (θ (k)) x (k) + B (θ (k)) u (k), \end{matrix}$ where $k \in Z_{+}$ , $x (k) \in R^{n}$ is the system state and $u (k) \in R^{m}$ is the control input; ${θ (k) : k \in Z_{+}}$ denotes the system mode which is a time-homogeneous Markovian process with right continuous trajectories. It is assumed that ${θ (k) : k \in Z_{+}}$ takes values on the finite set $ℑ ≜ {1, 2, \dots, η}$ with transition probability matrix $Π_{1} ≜ [$

Main results

The following lemma is firstly introduced, which is useful for the development of our work, the proof of which can be seen in Xiong and Lam (2006).

Lemma 1

Xiong & Lam, 2006

Given $d \in N$ , we define two sets $ℑ^{d + 1} ≜ ℑ \times ℑ \times \dots \times ℑ$ and $ℑ_{d + 1} ≜ {1, 2, \dots, η^{d + 1}}$ , and introduce the mapping $ψ : ℑ^{d + 1} \to ℑ_{d + 1}$ with $ψ (χ) = i + (i_{- 1} - 1) η + \dots + (i_{- d + 1} - 1) η^{d - 1} + (i_{- d} - 1) η^{d}$ where $χ = {[i i_{- 1} i_{- 2} \dots i_{- d}]}^{T} \in ℑ^{d + 1}$ and $i, i_{- 1}, \dots, i_{- d} \in ℑ$ . Then, the mapping $ψ (\cdot)$ is a bijection from $ℑ^{d + 1}$ to $ℑ_{d + 1}$ .

Theorem 1

Closed-loop system(10)is a delay-free Markovian jump linear system possessing $d η^{d + 1}$ operation modes with two

Numerical example

In this section, for the purpose of illustrating the usefulness and flexibility of the theory developed in this paper, we present a simulation example. Attention is focused on the design of a mode-dependent stabilizing controller for a Markovian jump linear system. We assume that $θ_{k} \in ℑ ≜ {1, 2}$ , that is $η = 2$ . Also, we assume that the random-network-induced delay $τ_{k} \in ℵ ≜ {1, 2}$ , that is $d = 2$ . Thus we have $ℑ_{3} = {1, 2, 3, 4, 5, 6, 7, 8}$ , $ℑ^{3} = {[\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}], [\begin{matrix} 1 \\ 1 \\ 2 \end{matrix}], [\begin{matrix} 2 \\ 1 \\ 1 \end{matrix}], [\begin{matrix} 2 \\ 1 \\ 2 \end{matrix}], [\begin{matrix} 1 \\ 2 \\ 1 \end{matrix}], [\begin{matrix} 1 \\ 2 \\ 2 \end{matrix}], [\begin{matrix} 2 \\ 2 \\ 1 \end{matrix}], [\begin{matrix} 2 \\ 2 \\ 2 \end{matrix}]} .$

By Theorem 1, system (10)

Conclusion

In this paper, we study the stabilization problem for a class of networked control systems with a discrete-time Markovian jump linear plant. Based on the approach of Xiong and Lam (2006), the resulting closed-loop system is modeled as a Markovian jump linear system with two Markovian jump parameters. A necessary and sufficient conditions of stochastic stability for NCSs is obtained in terms of a set of LMIs with matrix inversion constraints, from which the state-feedback gain can be solved by

Acknowledgements

The authors thank the referees and the editor for their valuable comments and suggestions.

Ming Liu earned his B.S. and M.S. degrees from Northeastern University, Shenyang, China, in 2003 and 2006, respectively. He is now pursuing his Ph.D. degree in the Department of Mathematics at City University of Hong Kong. His current research interests include quantized control and networked control systems.

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Daniel W.C. Ho received first class B.Sc., M.Sc. and Ph.D. degrees in mathematics from the University of Salford (UK) in 1980, 1982 and 1986, respectively. From 1985 to 1988, Dr. Ho was a Research Fellow in Industrial Control Unit, University of Strathclyde (Glasgow, Scotland). In 1989, he joined the Department of Mathematics, City University of Hong Kong, where he is currently a Professor. He is now serving as an Associate Editor for Asian Journal of Control. His research interests include H-infinity control theory, adaptive neural wavelet identification, nonlinear control theory, complex network, networked control system and quantized control.

Yugang Niu received the B.Sc. degree from Hebei Normal University, Shijiazhuang, PR China, in 1986, and the M.Sc. and Ph.D. degrees from the Nanjing University of Science & Technology, Nanjing, PR China, in 1992 and 2001, respectively. His postdoctoral research was carried out in the East China University of Science & Technology, Shanghai, PR China, from May 2001 to June 2003. In 2002, as Research Associate, he visited the University of Hong Kong for three months. In 2005 and 2006, as Research Fellow, he visited the City University of Hong Kong for six months, respectively.

In 2001, Dr. Niu joined the Department of Automation, the East China University of Science & Technology, where he is currently a Professor.

Dr. Niu has published more than 10 papers in the past 5 years in Automatica, IEEE Trans. Automatic Control, Systems & Control Letters, IEEE Trans. Fuzzy Systems, IEE Proc. Control Theory and Application and so on. His research interests include sliding mode control, stochastic systems, networked control systems, Markovian jumping systems, filtering, etc.

^☆: This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor George Yin under the direction of Editor Ian R. Petersen. This work was supported by CityU SRG 7002208, and the China National Natural Science Foundation 60674015, and Shanghai Leading Academic Discipline Project (B504).

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Brief paperStabilization of Markovian jump linear system over networks with random communication delay☆

Abstract

Introduction

Section snippets

Problem formulation

Main results

Xiong & Lam, 2006

Numerical example

Conclusion

Acknowledgements

Journal of the Franklin Institute

Systems and Control Letters

Automatica

Automatica

Automatica

Automatica

Brief paper
Stabilization of Markovian jump linear system over networks with random communication delay☆